Linear-Spaces.md

Linear SpacesLinear (Vector) SpacesDefinition: A linear space (or vector space) over a field FFF is a set VVV with two operations, addition and scalar multiplication such that the following axioms hold:
Vector Addition (+):V×V→V(+): V \times V \rightarrow V(+):V×V→V such that x+y∈Vx + y \in Vx+y∈V for all x,y∈Vx,y \in Vx,y∈V.
(A1) x+y=y+xx + y = y + xx+y=y+x   ∀x,y∈V\forall x,y \in V∀x,y∈V (commutativity)
(A2) x+(y+z)=(x+y)+zx + (y + z) = (x + y) + zx+(y+z)=(x+y)+z   ∀x,y,z∈V\forall x,y,z \in V∀x,y,z∈V (associativity)
(A3) There exists an element 0∈V0 \in V0∈V such that x+0=xx + 0 = xx+0=x   ∀x∈V\forall x \in V∀x∈V (existence of additive identity)
(A4) For every x∈Vx \in Vx∈V there exist an element −x∈V-x \in V−x∈V such that x+(−x)=0x + (-x) = 0x+(−x)=0   ∀x∈V\forall x \in V∀x∈V (existence of additive inverse)
Scalar Multiplication (⋅):F×V→V(\cdot): F \times V \rightarrow V(⋅):F×V→V such that αx∈V\alpha x \in Vαx∈V for all α∈F\alpha \in Fα∈F and x∈Vx \in Vx∈V.
(M1) α(bx)=(αb)x\alpha (bx) = (\alpha b)xα(bx)=(αb)x   ∀α,β∈F\forall \alpha, \beta \in F∀α,β∈F and ∀x∈V\forall x \in V∀x∈V (associativity)
(M2) α(x+y)=αx+αy\alpha(x+y) = \alpha x + \alpha yα(x+y)=αx+αy   ∀α∈F\forall \alpha \in F∀α∈F and ∀x,y∈V\forall x,y \in V∀x,y∈V (distributivity)
(M3) (α+β)x=αx+βx(\alpha + \beta)x = \alpha x + \beta x(α+β)x=αx+βx   ∀α,β∈F\forall \alpha, \beta \in F∀α,β∈F and ∀x∈V\forall x \in V∀x∈V (distributivity)
(M4) 1Fx=x1_Fx = x1F​x=x   ∀x∈V\forall x \in V∀x∈V (existence of multiplicative identity)
Elements of the vector space are called vectors, or points.
Scalar multiplication in a vector space depends on the field FFF. Thus when we say x∈Vx \in Vx∈V, we mean that xxx is a vector in the vector space VVV over the field FFF or (V,F)(V,F)(V,F).

For example, Rn\mathbb{R}^nRn is a vector space over the field R\mathbb{R}R, and Cn\mathbb{C}^nCn is a vector space over the field C\mathbb{C}C.
Sharing the same field is a necessary condition for two vector spaces to be comparable. For example, Rn\mathbb{R}^nRn and Cn\mathbb{C}^nCn are not comparable.
The simplest vector space contains only one point. In other words, {0}\{0\}{0} is a vector space.
Example: Show that x0F=0Vx 0_F  = 0_Vx0F​=0V​ for all x∈Vx \in Vx∈V

Proof:
y=x0F=0Vy = x0_F = 0_Vy=x0F​=0V​ is assumed to be true.
x+y=x+x0F=x1F+x0Fx + y = x + x0_F = x1_F+ x0_Fx+y=x+x0F​=x1F​+x0F​      (M4)
x+y=x(1F+0F)=x1Fx + y = x(1_F + 0_F) = x1_Fx+y=x(1F​+0F​)=x1F​            (M3)
x+y+(−x)=x1F+(−x1F)x + y + (-x) = x1_F + (-x1_F)x+y+(−x)=x1F​+(−x1F​)      (A4)
y=x(1F+(−1F))=x0V=0Vy = x(1_F + (-1_F)) = x0_V = 0_Vy=x(1F​+(−1F​))=x0V​=0V​     (A2)    ■\blacksquare■
Example: Show that x0F=0vx0_F = 0_vx0F​=0v​

Proof:
Assume y=x0Fy = x0_Fy=x0F​ and show y=0Vy = 0_Vy=0V​.
x+y=x+x0F=x1F+x0Fx + y = x + x0_F = x1_F + x0_Fx+y=x+x0F​=x1F​+x0F​       (M4)
x+y=x1F+x0F=x(1F+0F)x + y = x1_F + x0_F = x(1_F + 0_F)x+y=x1F​+x0F​=x(1F​+0F​)   (M3)
x+y=x(1F+0F)=x1Fx + y = x(1_F + 0_F) = x1_Fx+y=x(1F​+0F​)=x1F​              (A3)
x+y=x1F=xx + y = x1_F = xx+y=x1F​=x                            (M4)
x+y+(−x)=x+(−x)x + y + (-x) = x + (-x)x+y+(−x)=x+(−x)               (A4)
y=x+(−x)=0Vy = x + (-x) = 0_Vy=x+(−x)=0V​                         (A4)    ■\blacksquare■
Remark: A vector space has a unique additive identity.
Function SpaceDefinition: Let SSS be a set and FFF be a field. The set of all functions from SSS to FFF is denoted by FSF^SFS.
For f,g∈FSf,g \in F^Sf,g∈FS, the sum f+g∈FSf+g \in F^Sf+g∈FS is the function defined by
(f+g)(x)=f(x)+g(x), ∀x∈S(f+g)(x) = f(x) + g(x), \ \forall x \in S(f+g)(x)=f(x)+g(x), ∀x∈SFor f∈FSf \in F^Sf∈FS and α∈F\alpha \in Fα∈F, the product αf∈FS\alpha f \in F^Sαf∈FS is the function defined by
(αf)(x)=αf(x), ∀x∈S(\alpha f)(x) = \alpha f(x), \ \forall x \in S(αf)(x)=αf(x), ∀x∈SExample: Set of all polynomials with degree n with coefficients in FFF is denoted by F[x]nF[x]_nF[x]n​.
What essentially function spaces are, is that they are the set of all functions from a set to a field. For example, the set of all polynomials with degree n with coefficients in FFF is denoted by F[x]nF[x]_nF[x]n​.
#EE501 - Linear Systems Theory at METU